#14407 ⟨a, b | aaaa=b, bbbbb=a⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality

Properties

Commutative structure

Finite structure

An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19aa2
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19aa2a3
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19aa2a3a4
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19aa2a3a4a5
a6a6a7a8a9a10a11a12a13a14a15a16a17a18a19aa2a3a4a5a6
a7a7a8a9a10a11a12a13a14a15a16a17a18a19aa2a3a4a5a6a7
a8a8a9a10a11a12a13a14a15a16a17a18a19aa2a3a4a5a6a7a8
a9a9a10a11a12a13a14a15a16a17a18a19aa2a3a4a5a6a7a8a9
a10a10a11a12a13a14a15a16a17a18a19aa2a3a4a5a6a7a8a9a10
a11a11a12a13a14a15a16a17a18a19aa2a3a4a5a6a7a8a9a10a11
a12a12a13a14a15a16a17a18a19aa2a3a4a5a6a7a8a9a10a11a12
a13a13a14a15a16a17a18a19aa2a3a4a5a6a7a8a9a10a11a12a13
a14a14a15a16a17a18a19aa2a3a4a5a6a7a8a9a10a11a12a13a14
a15a15a16a17a18a19aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a16a16a17a18a19aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
a17a17a18a19aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
a18a18a19aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
a19a19aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19

Right Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a20 ⇒ a [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,bbbbb=a a/b - -
aaaaaaaaaaaaaaaaaaaa=a
b=aaaa

Same cardinality

20 unique, 65 total

Σ#PresentationPropertiesDescriptionφ
8526⟨a, b | aab=a, bbbb=1⟩Noc 2019
92207⟨a, b | ab=aa, bbbb=b⟩Noc 201
104095⟨a, b | baa=abb, abab=1⟩Grp 205
104212⟨a, b | aaaaa=1, abbbb=1⟩Grp Com 20ℤ2016
105349⟨a, b | aaa=bb, bab=aa⟩Noc 201
106728⟨a, b | aba=a, aaaa=bb⟩Noc 20
106764⟨a, b | aba=b, aaaa=bb⟩Noc 20
107117⟨a, b | ab=aa, bbbb=bb⟩Noc 20
1112159⟨a, b | aaaa=aa, abbb=b⟩Noc 20
1112322⟨a, b | aaab=bb, bbba=a⟩Noc 20
1114367⟨a, b | aaaa=a, bbbbb=a⟩Noc Com 20ℕ(20 = 5)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Noc Com 20ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Noc 20
1116124⟨a, b | aab=aa, baba=bb⟩Noc 20
1116339⟨a, b | aba=aa, aaaa=bb⟩Noc 20
1116343⟨a, b | aba=aa, aaab=bb⟩Noc 20
1116545⟨a, b | aba=bb, abb=aaa⟩Noc 20
1118619⟨a, b | aba=b, aaaaabb=1⟩Grp 203
1119503⟨a, b | aab=a, bbbbb=bb⟩Noc 20
1121047⟨a, b | ab=aa, bbbb=bbb⟩Noc 20